Euthanasia, (from Greek: åõèáíáóßá -åõ, eu, "good", èÜíáôïò, thanatos, death),is the practice of terminating the life of a person or animal in a painless or minimally painful way in order to stop suffering or other undesired conditions in life.

So in fact it's just ending someone's misery.

The man could have lived for another month, but the cancer had gone too far. They performed euthanasia to end his life withoug further pain.

Assisted suicide from a doctor becuase the patient is terminally ill and wishes to die. A lot of guidlines are set before a doctor can declare it acceptable to kill this person on thier behalf.
So far oregon is the only state to allow Euthanasia to legally take place.

The woman was termanilly ill and wanted to speed the process of her death up. After the doctor reviewed her case and stuff, he decided that he would proced with euthanasia if thats what she needed.